Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 92

Chapter 3, Problem 92

What is the relationship between the graphs of ƒ(x)=|x| and g(x)=|-x|?

Verified step by step guidance

Recall the definition of the absolute value function: for any real number $x$, $|x|$ represents the distance of $x$ from zero on the number line, which is always non-negative.

Write down the two functions explicitly: $f(x) = |x|$ and $g(x) = |-x|$.

Use the property of absolute value that states $|a| = |-a|$ for any real number $a$. Applying this to $g(x)$, we get $g(x) = |-x| = |x|$.

Conclude that since $g(x)$ simplifies to $|x|$, the graphs of $f(x)$ and $g(x)$ are identical.

Therefore, the relationship between the graphs of $f(x)$ and $g(x)$ is that they coincide exactly; they are the same graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Function

The absolute value function, denoted |x|, outputs the non-negative value of x regardless of its sign. It creates a V-shaped graph symmetric about the y-axis, reflecting all negative inputs as positive outputs.

Function Transformation and Symmetry

Understanding how transformations affect graphs is key. Replacing x with -x reflects the graph across the y-axis. Since |x| is symmetric about the y-axis, this transformation does not change the graph's shape or position.

Graph Comparison

Comparing graphs involves analyzing their shapes, positions, and symmetries. Since ƒ(x) = |x| and g(x) = |-x| produce identical outputs for all x, their graphs coincide, illustrating that g(x) is essentially the same function as ƒ(x).

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Related Practice

Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these.

$y^{3} = x + 4$

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Textbook Question

Each of the following graphs is obtained from the graph of ƒ(x)=|x| or g(x)=√x by applying several of the transformations discussed in this section. Describe the transformations and give an equation for the graph.

Textbook Question

Given functions f and g, find (a)(ƒ∘g)(x) and its domain, and (b)(g∘ƒ)(x) and its domain. ƒ(x)=1/(x+4), g(x)=-(1/x)

Textbook Question

Describe how the graph of each function can be obtained from the graph of ƒ(x) = |x|. g(x) = -|x|

Textbook Question

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. |x| = |y|

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Textbook Question